Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.4% → 92.2%

Time: 32.1s

Alternatives: 18

Speedup: 28.1×

• 27.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 92.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -102000 \lor \neg \left(t \leq 8.2 \cdot 10^{-86}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{k}}{t \cdot \sin k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -102000.0) (not (<= t 8.2e-86)))
   (/
    (pow
     (* (/ (cbrt (/ 2.0 (tan k))) t) (* (cbrt l) (/ (cbrt l) (cbrt (sin k)))))
     3.0)
    (+ 2.0 (pow (/ k t) 2.0)))
   (* (/ l (sin k)) (* 2.0 (/ (* (/ l k) (/ (cos k) k)) (* t (sin k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -102000.0) || !(t <= 8.2e-86)) {
		tmp = pow(((cbrt((2.0 / tan(k))) / t) * (cbrt(l) * (cbrt(l) / cbrt(sin(k))))), 3.0) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = (l / sin(k)) * (2.0 * (((l / k) * (cos(k) / k)) / (t * sin(k))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -102000.0) || !(t <= 8.2e-86)) {
		tmp = Math.pow(((Math.cbrt((2.0 / Math.tan(k))) / t) * (Math.cbrt(l) * (Math.cbrt(l) / Math.cbrt(Math.sin(k))))), 3.0) / (2.0 + Math.pow((k / t), 2.0));
	} else {
		tmp = (l / Math.sin(k)) * (2.0 * (((l / k) * (Math.cos(k) / k)) / (t * Math.sin(k))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -102000.0) || !(t <= 8.2e-86))
		tmp = Float64((Float64(Float64(cbrt(Float64(2.0 / tan(k))) / t) * Float64(cbrt(l) * Float64(cbrt(l) / cbrt(sin(k))))) ^ 3.0) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(Float64(l / sin(k)) * Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(cos(k) / k)) / Float64(t * sin(k)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -102000.0], N[Not[LessEqual[t, 8.2e-86]], $MachinePrecision]], N[(N[Power[N[(N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision] * N[(N[Power[l, 1/3], $MachinePrecision] * N[(N[Power[l, 1/3], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -102000 or 8.19999999999999959e-86 < t

   1. Initial program 69.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 69.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 58.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 58.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 69.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 69.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 69.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 69.6%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 70.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 70.1%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 69.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 69.5%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 77.9%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 77.9%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 77.9%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 78.0%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 77.9%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 77.9%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. cbrt-prod 91.2%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 91.2%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   10. Step-by-step derivation

       1. cbrt-div 94.9%

          \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\color{blue}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}} \cdot \sqrt[3]{\ell}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   11. Applied egg-rr 94.9%

       \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\color{blue}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}} \cdot \sqrt[3]{\ell}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   if -102000 < t < 8.19999999999999959e-86

   1. Initial program 40.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 40.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 40.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 40.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 40.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 40.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 40.3%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 40.3%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 40.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 40.3%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 40.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 20.9%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-udef 20.1%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*l/ 20.1%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-*r* 20.1%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 20.1%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. expm1-def 20.9%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-log1p 40.3%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*r* 40.3%

         \[\leadsto \frac{\frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. times-frac 43.5%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{\tan k \cdot {t}^{3}} \cdot \frac{\ell}{\sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      5. *-commutative 43.5%

         \[\leadsto \frac{\frac{2 \cdot \ell}{\color{blue}{{t}^{3} \cdot \tan k}} \cdot \frac{\ell}{\sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 43.5%

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{{t}^{3} \cdot \tan k} \cdot \frac{\ell}{\sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. div-inv 43.4%

         \[\leadsto \color{blue}{\left(\frac{2 \cdot \ell}{{t}^{3} \cdot \tan k} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      2. *-commutative 43.4%

         \[\leadsto \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{2 \cdot \ell}{{t}^{3} \cdot \tan k}\right)} \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. times-frac 44.7%

         \[\leadsto \left(\frac{\ell}{\sin k} \cdot \color{blue}{\left(\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}\right)}\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 44.7%

      \[\leadsto \color{blue}{\left(\frac{\ell}{\sin k} \cdot \left(\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}\right)\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. associate-*l* 45.5%

          \[\leadsto \color{blue}{\frac{\ell}{\sin k} \cdot \left(\left(\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)} \]

       2. associate-*r/ 45.5%

          \[\leadsto \frac{\ell}{\sin k} \cdot \color{blue}{\frac{\left(\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot 1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

       3. *-rgt-identity 45.5%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       4. associate-*r/ 45.5%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \ell}{\tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       5. associate-*l/ 46.2%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\frac{\color{blue}{\frac{2 \cdot \ell}{{t}^{3}}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       6. *-commutative 46.2%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot 2}}{{t}^{3}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       7. associate-/r* 44.3%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\color{blue}{\frac{\ell \cdot 2}{{t}^{3} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   11. Simplified 44.3%

       \[\leadsto \color{blue}{\frac{\ell}{\sin k} \cdot \frac{\frac{\ell \cdot 2}{{t}^{3} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   12. Taylor expanded in t around 0 87.0%

       \[\leadsto \frac{\ell}{\sin k} \cdot \color{blue}{\left(2 \cdot \frac{\ell \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right)} \]
   13. Step-by-step derivation

       1. associate-/r* 85.6%

          \[\leadsto \frac{\ell}{\sin k} \cdot \left(2 \cdot \color{blue}{\frac{\frac{\ell \cdot \cos k}{{k}^{2}}}{t \cdot \sin k}}\right) \]

       2. unpow2 85.6%

          \[\leadsto \frac{\ell}{\sin k} \cdot \left(2 \cdot \frac{\frac{\ell \cdot \cos k}{\color{blue}{k \cdot k}}}{t \cdot \sin k}\right) \]

       3. times-frac 90.6%

          \[\leadsto \frac{\ell}{\sin k} \cdot \left(2 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\cos k}{k}}}{t \cdot \sin k}\right) \]

   14. Simplified 90.6%

       \[\leadsto \frac{\ell}{\sin k} \cdot \color{blue}{\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{k}}{t \cdot \sin k}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -102000 \lor \neg \left(t \leq 8.2 \cdot 10^{-86}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{k}}{t \cdot \sin k}\right)\\ \end{array} \]

Alternative 2: 87.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(t_2 + 1\right)\right)} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{t_1}\right)\right)}^{3}}{2 + t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{k}}{t \cdot \sin k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (sin k))) (t_2 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))
          (+ 1.0 (+ t_2 1.0))))
        2e+305)
     (/
      (pow (* (/ (cbrt (/ 2.0 (tan k))) t) (* (cbrt l) (cbrt t_1))) 3.0)
      (+ 2.0 t_2))
     (* t_1 (* 2.0 (/ (* (/ l k) (/ (cos k) k)) (* t (sin k))))))))

C

double code(double t, double l, double k) {
	double t_1 = l / sin(k);
	double t_2 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))) * (1.0 + (t_2 + 1.0)))) <= 2e+305) {
		tmp = pow(((cbrt((2.0 / tan(k))) / t) * (cbrt(l) * cbrt(t_1))), 3.0) / (2.0 + t_2);
	} else {
		tmp = t_1 * (2.0 * (((l / k) * (cos(k) / k)) / (t * sin(k))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = l / Math.sin(k);
	double t_2 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))) * (1.0 + (t_2 + 1.0)))) <= 2e+305) {
		tmp = Math.pow(((Math.cbrt((2.0 / Math.tan(k))) / t) * (Math.cbrt(l) * Math.cbrt(t_1))), 3.0) / (2.0 + t_2);
	} else {
		tmp = t_1 * (2.0 * (((l / k) * (Math.cos(k) / k)) / (t * Math.sin(k))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(l / sin(k))
	t_2 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))) * Float64(1.0 + Float64(t_2 + 1.0)))) <= 2e+305)
		tmp = Float64((Float64(Float64(cbrt(Float64(2.0 / tan(k))) / t) * Float64(cbrt(l) * cbrt(t_1))) ^ 3.0) / Float64(2.0 + t_2));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(cos(k) / k)) / Float64(t * sin(k)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+305], N[(N[Power[N[(N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision] * N[(N[Power[l, 1/3], $MachinePrecision] * N[Power[t$95$1, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 1.9999999999999999e305

   1. Initial program 82.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 82.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 71.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 71.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 82.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 82.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 82.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 82.7%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 83.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 83.2%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 82.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 82.6%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 91.0%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 91.0%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 91.0%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 91.2%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 91.0%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 91.0%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. cbrt-prod 94.7%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 94.7%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   if 1.9999999999999999e305 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 22.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 22.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 22.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 22.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 22.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 22.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 22.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 22.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 22.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 22.7%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 22.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 22.7%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-udef 22.7%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*l/ 22.7%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-*r* 22.7%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 22.7%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. expm1-def 22.7%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-log1p 22.7%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*r* 22.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. times-frac 32.4%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{\tan k \cdot {t}^{3}} \cdot \frac{\ell}{\sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      5. *-commutative 32.4%

         \[\leadsto \frac{\frac{2 \cdot \ell}{\color{blue}{{t}^{3} \cdot \tan k}} \cdot \frac{\ell}{\sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 32.4%

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{{t}^{3} \cdot \tan k} \cdot \frac{\ell}{\sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. div-inv 32.4%

         \[\leadsto \color{blue}{\left(\frac{2 \cdot \ell}{{t}^{3} \cdot \tan k} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      2. *-commutative 32.4%

         \[\leadsto \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{2 \cdot \ell}{{t}^{3} \cdot \tan k}\right)} \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. times-frac 32.4%

         \[\leadsto \left(\frac{\ell}{\sin k} \cdot \color{blue}{\left(\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}\right)}\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 32.4%

      \[\leadsto \color{blue}{\left(\frac{\ell}{\sin k} \cdot \left(\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}\right)\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. associate-*l* 34.1%

          \[\leadsto \color{blue}{\frac{\ell}{\sin k} \cdot \left(\left(\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)} \]

       2. associate-*r/ 34.1%

          \[\leadsto \frac{\ell}{\sin k} \cdot \color{blue}{\frac{\left(\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot 1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

       3. *-rgt-identity 34.1%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       4. associate-*r/ 34.1%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \ell}{\tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       5. associate-*l/ 34.1%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\frac{\color{blue}{\frac{2 \cdot \ell}{{t}^{3}}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       6. *-commutative 34.1%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot 2}}{{t}^{3}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       7. associate-/r* 34.1%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\color{blue}{\frac{\ell \cdot 2}{{t}^{3} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   11. Simplified 34.1%

       \[\leadsto \color{blue}{\frac{\ell}{\sin k} \cdot \frac{\frac{\ell \cdot 2}{{t}^{3} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   12. Taylor expanded in t around 0 78.7%

       \[\leadsto \frac{\ell}{\sin k} \cdot \color{blue}{\left(2 \cdot \frac{\ell \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right)} \]
   13. Step-by-step derivation

       1. associate-/r* 75.7%

          \[\leadsto \frac{\ell}{\sin k} \cdot \left(2 \cdot \color{blue}{\frac{\frac{\ell \cdot \cos k}{{k}^{2}}}{t \cdot \sin k}}\right) \]

       2. unpow2 75.7%

          \[\leadsto \frac{\ell}{\sin k} \cdot \left(2 \cdot \frac{\frac{\ell \cdot \cos k}{\color{blue}{k \cdot k}}}{t \cdot \sin k}\right) \]

       3. times-frac 83.5%

          \[\leadsto \frac{\ell}{\sin k} \cdot \left(2 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\cos k}{k}}}{t \cdot \sin k}\right) \]

   14. Simplified 83.5%

       \[\leadsto \frac{\ell}{\sin k} \cdot \color{blue}{\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{k}}{t \cdot \sin k}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{\ell}{\sin k}}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{k}}{t \cdot \sin k}\right)\\ \end{array} \]

Alternative 3: 72.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-38}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 7.4 \cdot 10^{+148}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(2 \cdot \frac{\ell \cdot \cos k}{\left(t \cdot \sin k\right) \cdot {k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 7e-38)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (if (<= k 7.4e+148)
     (* (/ l (sin k)) (* 2.0 (/ (* l (cos k)) (* (* t (sin k)) (pow k 2.0)))))
     (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t (pow (sin k) 2.0))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 7e-38) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else if (k <= 7.4e+148) {
		tmp = (l / sin(k)) * (2.0 * ((l * cos(k)) / ((t * sin(k)) * pow(k, 2.0))));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7d-38) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else if (k <= 7.4d+148) then
        tmp = (l / sin(k)) * (2.0d0 * ((l * cos(k)) / ((t * sin(k)) * (k ** 2.0d0))))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 7e-38) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else if (k <= 7.4e+148) {
		tmp = (l / Math.sin(k)) * (2.0 * ((l * Math.cos(k)) / ((t * Math.sin(k)) * Math.pow(k, 2.0))));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 7e-38:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	elif k <= 7.4e+148:
		tmp = (l / math.sin(k)) * (2.0 * ((l * math.cos(k)) / ((t * math.sin(k)) * math.pow(k, 2.0))))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 7e-38)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	elseif (k <= 7.4e+148)
		tmp = Float64(Float64(l / sin(k)) * Float64(2.0 * Float64(Float64(l * cos(k)) / Float64(Float64(t * sin(k)) * (k ^ 2.0)))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 7e-38)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	elseif (k <= 7.4e+148)
		tmp = (l / sin(k)) * (2.0 * ((l * cos(k)) / ((t * sin(k)) * (k ^ 2.0))));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 7e-38], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7.4e+148], N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 7.0000000000000003e-38

   1. Initial program 61.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.2%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 61.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 61.9%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 62.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 62.2%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 61.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 61.8%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 74.8%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 74.8%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 74.8%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 74.9%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 74.8%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 74.8%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Taylor expanded in k around 0 53.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   9. Step-by-step derivation

      1. associate-/l/ 53.9%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      2. unpow2 53.9%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}{{k}^{2}} \]

      3. associate-*l/ 57.6%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{{t}^{3}} \cdot \ell}}{{k}^{2}} \]

      4. unpow2 57.6%

         \[\leadsto \frac{\frac{\ell}{{t}^{3}} \cdot \ell}{\color{blue}{k \cdot k}} \]

      5. times-frac 69.8%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

      6. associate-/r* 67.9%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3} \cdot k}} \cdot \frac{\ell}{k} \]

      7. associate-/l/ 70.7%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \cdot \frac{\ell}{k} \]

   10. Simplified 70.7%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{\ell}{k}} \]

   if 7.0000000000000003e-38 < k < 7.4000000000000005e148

   1. Initial program 49.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 49.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 49.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 49.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 49.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 49.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 49.5%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 49.5%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 49.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 49.5%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 49.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 38.9%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-udef 36.5%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*l/ 36.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-*r* 36.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 36.5%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. expm1-def 38.9%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-log1p 49.5%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*r* 49.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. times-frac 56.7%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{\tan k \cdot {t}^{3}} \cdot \frac{\ell}{\sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      5. *-commutative 56.7%

         \[\leadsto \frac{\frac{2 \cdot \ell}{\color{blue}{{t}^{3} \cdot \tan k}} \cdot \frac{\ell}{\sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 56.7%

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{{t}^{3} \cdot \tan k} \cdot \frac{\ell}{\sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. div-inv 56.7%

         \[\leadsto \color{blue}{\left(\frac{2 \cdot \ell}{{t}^{3} \cdot \tan k} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      2. *-commutative 56.7%

         \[\leadsto \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{2 \cdot \ell}{{t}^{3} \cdot \tan k}\right)} \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. times-frac 56.7%

         \[\leadsto \left(\frac{\ell}{\sin k} \cdot \color{blue}{\left(\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}\right)}\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 56.7%

      \[\leadsto \color{blue}{\left(\frac{\ell}{\sin k} \cdot \left(\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}\right)\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. associate-*l* 61.1%

          \[\leadsto \color{blue}{\frac{\ell}{\sin k} \cdot \left(\left(\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)} \]

       2. associate-*r/ 61.1%

          \[\leadsto \frac{\ell}{\sin k} \cdot \color{blue}{\frac{\left(\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot 1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

       3. *-rgt-identity 61.1%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       4. associate-*r/ 61.2%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \ell}{\tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       5. associate-*l/ 61.2%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\frac{\color{blue}{\frac{2 \cdot \ell}{{t}^{3}}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       6. *-commutative 61.2%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot 2}}{{t}^{3}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       7. associate-/r* 61.2%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\color{blue}{\frac{\ell \cdot 2}{{t}^{3} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   11. Simplified 61.2%

       \[\leadsto \color{blue}{\frac{\ell}{\sin k} \cdot \frac{\frac{\ell \cdot 2}{{t}^{3} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   12. Taylor expanded in t around 0 88.0%

       \[\leadsto \frac{\ell}{\sin k} \cdot \color{blue}{\left(2 \cdot \frac{\ell \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right)} \]

   if 7.4000000000000005e148 < k

   1. Initial program 34.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 34.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 34.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 34.4%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 34.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 34.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 34.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 34.4%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 34.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 34.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 34.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 34.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 47.3%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 47.3%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 47.3%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 47.3%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 47.3%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 47.3%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. cbrt-prod 62.2%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 62.2%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around inf 50.4%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   11. Step-by-step derivation

       1. times-frac 50.5%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

       2. unpow2 50.5%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

       3. unpow2 50.5%

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

       4. times-frac 95.1%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

   12. Simplified 95.1%

       \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-38}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 7.4 \cdot 10^{+148}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(2 \cdot \frac{\ell \cdot \cos k}{\left(t \cdot \sin k\right) \cdot {k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 4: 71.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-37}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 7 \cdot 10^{+139}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{k}}{t \cdot \sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.25e-37)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (if (<= k 7e+139)
     (* (/ l (sin k)) (* 2.0 (/ (* (/ l k) (/ (cos k) k)) (* t (sin k)))))
     (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t (pow (sin k) 2.0))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.25e-37) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else if (k <= 7e+139) {
		tmp = (l / sin(k)) * (2.0 * (((l / k) * (cos(k) / k)) / (t * sin(k))));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.25d-37) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else if (k <= 7d+139) then
        tmp = (l / sin(k)) * (2.0d0 * (((l / k) * (cos(k) / k)) / (t * sin(k))))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.25e-37) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else if (k <= 7e+139) {
		tmp = (l / Math.sin(k)) * (2.0 * (((l / k) * (Math.cos(k) / k)) / (t * Math.sin(k))));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.25e-37:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	elif k <= 7e+139:
		tmp = (l / math.sin(k)) * (2.0 * (((l / k) * (math.cos(k) / k)) / (t * math.sin(k))))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.25e-37)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	elseif (k <= 7e+139)
		tmp = Float64(Float64(l / sin(k)) * Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(cos(k) / k)) / Float64(t * sin(k)))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.25e-37)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	elseif (k <= 7e+139)
		tmp = (l / sin(k)) * (2.0 * (((l / k) * (cos(k) / k)) / (t * sin(k))));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.25e-37], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7e+139], N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.2499999999999999e-37

   1. Initial program 61.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.2%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 61.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 61.9%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 62.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 62.2%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 61.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 61.8%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 74.8%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 74.8%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 74.8%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 74.9%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 74.8%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 74.8%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Taylor expanded in k around 0 53.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   9. Step-by-step derivation

      1. associate-/l/ 53.9%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      2. unpow2 53.9%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}{{k}^{2}} \]

      3. associate-*l/ 57.6%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{{t}^{3}} \cdot \ell}}{{k}^{2}} \]

      4. unpow2 57.6%

         \[\leadsto \frac{\frac{\ell}{{t}^{3}} \cdot \ell}{\color{blue}{k \cdot k}} \]

      5. times-frac 69.8%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

      6. associate-/r* 67.9%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3} \cdot k}} \cdot \frac{\ell}{k} \]

      7. associate-/l/ 70.7%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \cdot \frac{\ell}{k} \]

   10. Simplified 70.7%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{\ell}{k}} \]

   if 1.2499999999999999e-37 < k < 6.99999999999999957e139

   1. Initial program 49.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 49.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 49.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 49.4%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 49.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 49.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 49.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 49.3%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 49.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 49.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 49.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 38.2%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-udef 35.7%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*l/ 35.7%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-*r* 35.7%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 35.7%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. expm1-def 38.2%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-log1p 49.4%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*r* 49.4%

         \[\leadsto \frac{\frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. times-frac 54.6%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{\tan k \cdot {t}^{3}} \cdot \frac{\ell}{\sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      5. *-commutative 54.6%

         \[\leadsto \frac{\frac{2 \cdot \ell}{\color{blue}{{t}^{3} \cdot \tan k}} \cdot \frac{\ell}{\sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 54.6%

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{{t}^{3} \cdot \tan k} \cdot \frac{\ell}{\sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. div-inv 54.5%

         \[\leadsto \color{blue}{\left(\frac{2 \cdot \ell}{{t}^{3} \cdot \tan k} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      2. *-commutative 54.5%

         \[\leadsto \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{2 \cdot \ell}{{t}^{3} \cdot \tan k}\right)} \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. times-frac 54.5%

         \[\leadsto \left(\frac{\ell}{\sin k} \cdot \color{blue}{\left(\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}\right)}\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 54.5%

      \[\leadsto \color{blue}{\left(\frac{\ell}{\sin k} \cdot \left(\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}\right)\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. associate-*l* 59.2%

          \[\leadsto \color{blue}{\frac{\ell}{\sin k} \cdot \left(\left(\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)} \]

       2. associate-*r/ 59.2%

          \[\leadsto \frac{\ell}{\sin k} \cdot \color{blue}{\frac{\left(\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot 1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

       3. *-rgt-identity 59.2%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       4. associate-*r/ 59.2%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \ell}{\tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       5. associate-*l/ 59.3%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\frac{\color{blue}{\frac{2 \cdot \ell}{{t}^{3}}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       6. *-commutative 59.3%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot 2}}{{t}^{3}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       7. associate-/r* 59.3%

          \[\leadsto \frac{\ell}{\sin k} \cdot \frac{\color{blue}{\frac{\ell \cdot 2}{{t}^{3} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   11. Simplified 59.3%

       \[\leadsto \color{blue}{\frac{\ell}{\sin k} \cdot \frac{\frac{\ell \cdot 2}{{t}^{3} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   12. Taylor expanded in t around 0 87.4%

       \[\leadsto \frac{\ell}{\sin k} \cdot \color{blue}{\left(2 \cdot \frac{\ell \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right)} \]
   13. Step-by-step derivation

       1. associate-/r* 79.6%

          \[\leadsto \frac{\ell}{\sin k} \cdot \left(2 \cdot \color{blue}{\frac{\frac{\ell \cdot \cos k}{{k}^{2}}}{t \cdot \sin k}}\right) \]

       2. unpow2 79.6%

          \[\leadsto \frac{\ell}{\sin k} \cdot \left(2 \cdot \frac{\frac{\ell \cdot \cos k}{\color{blue}{k \cdot k}}}{t \cdot \sin k}\right) \]

       3. times-frac 79.6%

          \[\leadsto \frac{\ell}{\sin k} \cdot \left(2 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\cos k}{k}}}{t \cdot \sin k}\right) \]

   14. Simplified 79.6%

       \[\leadsto \frac{\ell}{\sin k} \cdot \color{blue}{\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{k}}{t \cdot \sin k}\right)} \]

   if 6.99999999999999957e139 < k

   1. Initial program 35.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 35.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 35.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 35.4%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 35.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 35.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 35.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 35.4%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 35.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 35.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 35.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 35.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 47.6%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 47.6%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 47.6%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 47.6%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 47.6%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 47.6%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. cbrt-prod 64.3%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 64.3%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around inf 50.5%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   11. Step-by-step derivation

       1. times-frac 50.6%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

       2. unpow2 50.6%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

       3. unpow2 50.6%

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

       4. times-frac 95.4%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

   12. Simplified 95.4%

       \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-37}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 7 \cdot 10^{+139}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{k}}{t \cdot \sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 5: 71.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 1.25 \cdot 10^{-17}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+150}:\\ \;\;\;\;\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t}\right) \cdot \frac{2}{t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 1.25e-17)
     (* (/ l k) (/ (/ l k) (pow t 3.0)))
     (if (<= k 8.5e+150)
       (* (* (/ l (* k k)) (/ (* l (cos k)) t)) (/ 2.0 t_1))
       (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t t_1))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 1.25e-17) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else if (k <= 8.5e+150) {
		tmp = ((l / (k * k)) * ((l * cos(k)) / t)) * (2.0 / t_1);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    if (k <= 1.25d-17) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else if (k <= 8.5d+150) then
        tmp = ((l / (k * k)) * ((l * cos(k)) / t)) * (2.0d0 / t_1)
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (t * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 1.25e-17) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else if (k <= 8.5e+150) {
		tmp = ((l / (k * k)) * ((l * Math.cos(k)) / t)) * (2.0 / t_1);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * t_1)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 1.25e-17:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	elif k <= 8.5e+150:
		tmp = ((l / (k * k)) * ((l * math.cos(k)) / t)) * (2.0 / t_1)
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * t_1)))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 1.25e-17)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	elseif (k <= 8.5e+150)
		tmp = Float64(Float64(Float64(l / Float64(k * k)) * Float64(Float64(l * cos(k)) / t)) * Float64(2.0 / t_1));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 1.25e-17)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	elseif (k <= 8.5e+150)
		tmp = ((l / (k * k)) * ((l * cos(k)) / t)) * (2.0 / t_1);
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 1.25e-17], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 8.5e+150], N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.25e-17

   1. Initial program 61.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 61.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 61.7%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 62.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 62.1%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 61.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 61.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 74.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 74.5%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 74.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 74.6%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 74.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 74.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Taylor expanded in k around 0 53.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   9. Step-by-step derivation

      1. associate-/l/ 53.9%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      2. unpow2 53.9%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}{{k}^{2}} \]

      3. associate-*l/ 58.1%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{{t}^{3}} \cdot \ell}}{{k}^{2}} \]

      4. unpow2 58.1%

         \[\leadsto \frac{\frac{\ell}{{t}^{3}} \cdot \ell}{\color{blue}{k \cdot k}} \]

      5. times-frac 70.1%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

      6. associate-/r* 68.3%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3} \cdot k}} \cdot \frac{\ell}{k} \]

      7. associate-/l/ 71.0%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \cdot \frac{\ell}{k} \]

   10. Simplified 71.0%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{\ell}{k}} \]

   if 1.25e-17 < k < 8.4999999999999999e150

   1. Initial program 48.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 48.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 48.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 48.2%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 48.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 48.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 48.3%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 48.3%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 48.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 48.3%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 48.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 48.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 57.9%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 57.9%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 57.9%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 58.0%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 58.0%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 58.0%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. cbrt-prod 66.6%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 66.6%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around inf 74.7%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   11. Step-by-step derivation

       1. associate-*r/ 74.7%

          \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

       2. unpow2 74.7%

          \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       3. *-commutative 74.7%

          \[\leadsto \frac{\color{blue}{\left(\left(\ell \cdot \ell\right) \cdot \cos k\right) \cdot 2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       4. unpow2 74.7%

          \[\leadsto \frac{\left(\color{blue}{{\ell}^{2}} \cdot \cos k\right) \cdot 2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       5. associate-*r* 74.6%

          \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot 2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

       6. times-frac 74.7%

          \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot t} \cdot \frac{2}{{\sin k}^{2}}} \]

       7. unpow2 74.7%

          \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}{{k}^{2} \cdot t} \cdot \frac{2}{{\sin k}^{2}} \]

       8. associate-*l* 74.6%

          \[\leadsto \frac{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}{{k}^{2} \cdot t} \cdot \frac{2}{{\sin k}^{2}} \]

       9. times-frac 83.2%

          \[\leadsto \color{blue}{\left(\frac{\ell}{{k}^{2}} \cdot \frac{\ell \cdot \cos k}{t}\right)} \cdot \frac{2}{{\sin k}^{2}} \]

       10. unpow2 83.2%

           \[\leadsto \left(\frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell \cdot \cos k}{t}\right) \cdot \frac{2}{{\sin k}^{2}} \]

   12. Simplified 83.2%

       \[\leadsto \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t}\right) \cdot \frac{2}{{\sin k}^{2}}} \]

   if 8.4999999999999999e150 < k

   1. Initial program 34.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 34.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 34.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 34.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 34.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 34.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 34.5%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 34.5%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 34.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 34.5%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 34.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 34.5%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 45.2%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 45.2%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 45.2%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 45.2%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 45.2%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 45.2%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. cbrt-prod 58.4%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 58.4%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around inf 48.8%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   11. Step-by-step derivation

       1. times-frac 48.8%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

       2. unpow2 48.8%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

       3. unpow2 48.8%

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

       4. times-frac 94.6%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

   12. Simplified 94.6%

       \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-17}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+150}:\\ \;\;\;\;\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t}\right) \cdot \frac{2}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 6: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 9.6e-14)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (* 2.0 (* (* l (/ l (* k k))) (/ (/ (cos k) t) (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.6e-14) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 * ((l * (l / (k * k))) * ((cos(k) / t) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9.6d-14) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 * ((l * (l / (k * k))) * ((cos(k) / t) / (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.6e-14) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * ((l * (l / (k * k))) * ((Math.cos(k) / t) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 9.6e-14:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 * ((l * (l / (k * k))) * ((math.cos(k) / t) / math.pow(math.sin(k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 9.6e-14)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / Float64(k * k))) * Float64(Float64(cos(k) / t) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 9.6e-14)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 * ((l * (l / (k * k))) * ((cos(k) / t) / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 9.6e-14], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.599999999999999e-14

   1. Initial program 61.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 61.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 61.7%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 62.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 62.1%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 61.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 61.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 74.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 74.5%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 74.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 74.6%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 74.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 74.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Taylor expanded in k around 0 53.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   9. Step-by-step derivation

      1. associate-/l/ 53.9%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      2. unpow2 53.9%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}{{k}^{2}} \]

      3. associate-*l/ 58.1%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{{t}^{3}} \cdot \ell}}{{k}^{2}} \]

      4. unpow2 58.1%

         \[\leadsto \frac{\frac{\ell}{{t}^{3}} \cdot \ell}{\color{blue}{k \cdot k}} \]

      5. times-frac 70.1%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

      6. associate-/r* 68.3%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3} \cdot k}} \cdot \frac{\ell}{k} \]

      7. associate-/l/ 71.0%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \cdot \frac{\ell}{k} \]

   10. Simplified 71.0%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{\ell}{k}} \]

   if 9.599999999999999e-14 < k

   1. Initial program 42.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 42.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 42.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 42.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 42.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 42.6%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 42.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 64.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 64.1%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 64.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 64.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 64.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 68.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 68.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 68.2%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around inf 64.1%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   8. Step-by-step derivation

      1. times-frac 58.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

      2. unpow2 58.7%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

      3. associate-*r/ 71.3%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{{k}^{2}}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

      4. unpow2 71.3%

         \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

      5. associate-/r* 71.3%

         \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right) \]

   9. Simplified 71.3%

      \[\leadsto 2 \cdot \color{blue}{\left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 7: 71.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-15}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4e-15)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4e-15) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4d-15) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4e-15) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4e-15:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4e-15)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4e-15)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4e-15], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.0000000000000003e-15

   1. Initial program 61.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 61.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 61.7%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 62.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 62.1%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 61.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 61.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 74.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 74.5%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 74.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 74.6%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 74.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 74.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Taylor expanded in k around 0 53.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   9. Step-by-step derivation

      1. associate-/l/ 53.9%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      2. unpow2 53.9%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}{{k}^{2}} \]

      3. associate-*l/ 58.1%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{{t}^{3}} \cdot \ell}}{{k}^{2}} \]

      4. unpow2 58.1%

         \[\leadsto \frac{\frac{\ell}{{t}^{3}} \cdot \ell}{\color{blue}{k \cdot k}} \]

      5. times-frac 70.1%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

      6. associate-/r* 68.3%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3} \cdot k}} \cdot \frac{\ell}{k} \]

      7. associate-/l/ 71.0%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \cdot \frac{\ell}{k} \]

   10. Simplified 71.0%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{\ell}{k}} \]

   if 4.0000000000000003e-15 < k

   1. Initial program 42.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 42.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 42.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 42.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 42.6%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 42.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 42.6%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 42.5%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 52.7%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 52.7%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 52.7%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 52.8%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 52.8%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 52.8%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. cbrt-prod 63.3%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 63.3%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around inf 64.1%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   11. Step-by-step derivation

       1. times-frac 58.7%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

       2. unpow2 58.7%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

       3. unpow2 58.7%

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

       4. times-frac 84.0%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

   12. Simplified 84.0%

       \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-15}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 8: 72.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-74} \lor \neg \left(t \leq 3 \cdot 10^{-86}\right):\\ \;\;\;\;\frac{2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.4e-74) (not (<= t 3e-86)))
   (/ (* 2.0 (* (/ l k) (/ (/ l k) (pow t 3.0)))) (+ 2.0 (pow (/ k t) 2.0)))
   (*
    2.0
    (*
     (/ (cos k) (* k (* t k)))
     (fma 0.3333333333333333 (* l l) (* l (/ l (* k k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.4e-74) || !(t <= 3e-86)) {
		tmp = (2.0 * ((l / k) * ((l / k) / pow(t, 3.0)))) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = 2.0 * ((cos(k) / (k * (t * k))) * fma(0.3333333333333333, (l * l), (l * (l / (k * k)))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.4e-74) || !(t <= 3e-86))
		tmp = Float64(Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)))) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * Float64(t * k))) * fma(0.3333333333333333, Float64(l * l), Float64(l * Float64(l / Float64(k * k))))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1.4e-74], N[Not[LessEqual[t, 3e-86]], $MachinePrecision]], N[(N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 * N[(l * l), $MachinePrecision] + N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.39999999999999994e-74 or 3.0000000000000001e-86 < t

   1. Initial program 70.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 70.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 60.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 60.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 70.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 70.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 70.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 70.8%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 69.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 69.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 59.8%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-udef 57.4%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*l/ 57.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-*r* 57.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 57.4%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. expm1-def 59.8%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-log1p 69.8%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*r* 69.8%

         \[\leadsto \frac{\frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. times-frac 79.8%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{\tan k \cdot {t}^{3}} \cdot \frac{\ell}{\sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      5. *-commutative 79.8%

         \[\leadsto \frac{\frac{2 \cdot \ell}{\color{blue}{{t}^{3} \cdot \tan k}} \cdot \frac{\ell}{\sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 79.8%

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{{t}^{3} \cdot \tan k} \cdot \frac{\ell}{\sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Taylor expanded in k around 0 55.9%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   9. Step-by-step derivation

      1. associate-/l/ 56.0%

         \[\leadsto \frac{2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. unpow2 56.0%

         \[\leadsto \frac{2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}{{k}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*l/ 62.8%

         \[\leadsto \frac{2 \cdot \frac{\color{blue}{\frac{\ell}{{t}^{3}} \cdot \ell}}{{k}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. unpow2 62.8%

         \[\leadsto \frac{2 \cdot \frac{\frac{\ell}{{t}^{3}} \cdot \ell}{\color{blue}{k \cdot k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      5. times-frac 76.6%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      6. associate-/r* 74.9%

         \[\leadsto \frac{2 \cdot \left(\color{blue}{\frac{\ell}{{t}^{3} \cdot k}} \cdot \frac{\ell}{k}\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      7. associate-/l/ 77.6%

         \[\leadsto \frac{2 \cdot \left(\color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \cdot \frac{\ell}{k}\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Simplified 77.6%

       \[\leadsto \frac{\color{blue}{2 \cdot \left(\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{\ell}{k}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   if -1.39999999999999994e-74 < t < 3.0000000000000001e-86

   1. Initial program 34.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 34.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 34.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 34.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 34.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 34.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 34.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 34.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 34.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 34.0%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 34.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 79.9%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 79.9%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 79.9%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 80.8%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 80.8%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 85.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 85.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 85.6%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 71.2%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 71.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]

      2. unpow2 71.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, \frac{{\ell}^{2}}{{k}^{2}}\right)\right) \]

      3. unpow2 71.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right)\right) \]

      4. associate-*r/ 75.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \color{blue}{\ell \cdot \frac{\ell}{{k}^{2}}}\right)\right) \]

      5. unpow2 75.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right) \]

   9. Simplified 75.4%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{k \cdot k}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-74} \lor \neg \left(t \leq 3 \cdot 10^{-86}\right):\\ \;\;\;\;\frac{2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \]

Alternative 9: 65.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-37}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\cos k}{\frac{t}{0.3333333333333333 + \frac{1}{k \cdot k}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.1e-37)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (*
    2.0
    (*
     l
     (*
      (/ l (* k k))
      (/ (cos k) (/ t (+ 0.3333333333333333 (/ 1.0 (* k k))))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.1e-37) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 * (l * ((l / (k * k)) * (cos(k) / (t / (0.3333333333333333 + (1.0 / (k * k)))))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.1d-37) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 * (l * ((l / (k * k)) * (cos(k) / (t / (0.3333333333333333d0 + (1.0d0 / (k * k)))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.1e-37) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * (l * ((l / (k * k)) * (Math.cos(k) / (t / (0.3333333333333333 + (1.0 / (k * k)))))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.1e-37:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 * (l * ((l / (k * k)) * (math.cos(k) / (t / (0.3333333333333333 + (1.0 / (k * k)))))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.1e-37)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(l * Float64(Float64(l / Float64(k * k)) * Float64(cos(k) / Float64(t / Float64(0.3333333333333333 + Float64(1.0 / Float64(k * k))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.1e-37)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 * (l * ((l / (k * k)) * (cos(k) / (t / (0.3333333333333333 + (1.0 / (k * k)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.1e-37], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t / N[(0.3333333333333333 + N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.10000000000000001e-37

   1. Initial program 61.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.2%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 61.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 61.9%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 62.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 62.2%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 61.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 61.8%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 74.8%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 74.8%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 74.8%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 74.9%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 74.8%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 74.8%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Taylor expanded in k around 0 53.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   9. Step-by-step derivation

      1. associate-/l/ 53.9%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      2. unpow2 53.9%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}{{k}^{2}} \]

      3. associate-*l/ 57.6%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{{t}^{3}} \cdot \ell}}{{k}^{2}} \]

      4. unpow2 57.6%

         \[\leadsto \frac{\frac{\ell}{{t}^{3}} \cdot \ell}{\color{blue}{k \cdot k}} \]

      5. times-frac 69.8%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

      6. associate-/r* 67.9%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3} \cdot k}} \cdot \frac{\ell}{k} \]

      7. associate-/l/ 70.7%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \cdot \frac{\ell}{k} \]

   10. Simplified 70.7%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{\ell}{k}} \]

   if 1.10000000000000001e-37 < k

   1. Initial program 42.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 42.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 42.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 42.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 42.8%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 42.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 42.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 62.6%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 62.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 62.5%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 62.5%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 62.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 66.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 66.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 66.5%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 53.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 53.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]

      2. unpow2 53.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, \frac{{\ell}^{2}}{{k}^{2}}\right)\right) \]

      3. unpow2 53.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right)\right) \]

      4. associate-*r/ 55.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \color{blue}{\ell \cdot \frac{\ell}{{k}^{2}}}\right)\right) \]

      5. unpow2 55.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right) \]

   9. Simplified 55.2%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{k \cdot k}\right)}\right) \]

   10. Taylor expanded in l around 0 53.4%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)\right)}{{k}^{2} \cdot t}} \]
   11. Step-by-step derivation

       1. times-frac 50.8%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}{t}\right)} \]

       2. unpow2 50.8%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}{t}\right) \]

       3. associate-*r/ 57.4%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{{k}^{2}}\right)} \cdot \frac{\cos k \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}{t}\right) \]

       4. unpow2 57.4%

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right) \cdot \frac{\cos k \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}{t}\right) \]

       5. associate-*l* 59.0%

          \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\cos k \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}{t}\right)\right)} \]

       6. associate-/l* 59.0%

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\cos k}{\frac{t}{0.3333333333333333 + \frac{1}{{k}^{2}}}}}\right)\right) \]

       7. unpow2 59.0%

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\cos k}{\frac{t}{0.3333333333333333 + \frac{1}{\color{blue}{k \cdot k}}}}\right)\right) \]

   12. Simplified 59.0%

       \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\cos k}{\frac{t}{0.3333333333333333 + \frac{1}{k \cdot k}}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-37}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\cos k}{\frac{t}{0.3333333333333333 + \frac{1}{k \cdot k}}}\right)\right)\\ \end{array} \]

Alternative 10: 65.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-16}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 8 \cdot 10^{+85}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 7e-16)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (if (<= k 8e+85)
     (/ (* 2.0 (* l (* l (cos k)))) (* k (* (* k k) (* t k))))
     (* 2.0 (* 0.3333333333333333 (* (* (/ l k) (/ l k)) (/ (cos k) t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 7e-16) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else if (k <= 8e+85) {
		tmp = (2.0 * (l * (l * cos(k)))) / (k * ((k * k) * (t * k)));
	} else {
		tmp = 2.0 * (0.3333333333333333 * (((l / k) * (l / k)) * (cos(k) / t)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7d-16) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else if (k <= 8d+85) then
        tmp = (2.0d0 * (l * (l * cos(k)))) / (k * ((k * k) * (t * k)))
    else
        tmp = 2.0d0 * (0.3333333333333333d0 * (((l / k) * (l / k)) * (cos(k) / t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 7e-16) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else if (k <= 8e+85) {
		tmp = (2.0 * (l * (l * Math.cos(k)))) / (k * ((k * k) * (t * k)));
	} else {
		tmp = 2.0 * (0.3333333333333333 * (((l / k) * (l / k)) * (Math.cos(k) / t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 7e-16:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	elif k <= 8e+85:
		tmp = (2.0 * (l * (l * math.cos(k)))) / (k * ((k * k) * (t * k)))
	else:
		tmp = 2.0 * (0.3333333333333333 * (((l / k) * (l / k)) * (math.cos(k) / t)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 7e-16)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	elseif (k <= 8e+85)
		tmp = Float64(Float64(2.0 * Float64(l * Float64(l * cos(k)))) / Float64(k * Float64(Float64(k * k) * Float64(t * k))));
	else
		tmp = Float64(2.0 * Float64(0.3333333333333333 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 7e-16)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	elseif (k <= 8e+85)
		tmp = (2.0 * (l * (l * cos(k)))) / (k * ((k * k) * (t * k)));
	else
		tmp = 2.0 * (0.3333333333333333 * (((l / k) * (l / k)) * (cos(k) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 7e-16], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 8e+85], N[(N[(2.0 * N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(N[(k * k), $MachinePrecision] * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.3333333333333333 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 7.00000000000000035e-16

   1. Initial program 61.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 61.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 61.7%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 62.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 62.1%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 61.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 61.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 74.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 74.5%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 74.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 74.6%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 74.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 74.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Taylor expanded in k around 0 53.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   9. Step-by-step derivation

      1. associate-/l/ 53.9%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      2. unpow2 53.9%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}{{k}^{2}} \]

      3. associate-*l/ 58.1%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{{t}^{3}} \cdot \ell}}{{k}^{2}} \]

      4. unpow2 58.1%

         \[\leadsto \frac{\frac{\ell}{{t}^{3}} \cdot \ell}{\color{blue}{k \cdot k}} \]

      5. times-frac 70.1%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

      6. associate-/r* 68.3%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3} \cdot k}} \cdot \frac{\ell}{k} \]

      7. associate-/l/ 71.0%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \cdot \frac{\ell}{k} \]

   10. Simplified 71.0%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{\ell}{k}} \]

   if 7.00000000000000035e-16 < k < 8.0000000000000001e85

   1. Initial program 53.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 53.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 53.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 53.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 53.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 53.9%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 53.9%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 53.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 53.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 53.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 57.6%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 57.6%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 57.6%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 57.7%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 57.7%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 57.7%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Taylor expanded in k around inf 72.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 72.5%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. *-commutative 72.5%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\cos k \cdot {\ell}^{2}\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      3. unpow2 72.5%

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. associate-*r* 72.5%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. *-commutative 72.5%

         \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(\ell \cdot \cos k\right)} \cdot \ell\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. *-commutative 72.5%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-*r* 72.4%

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      8. unpow2 72.4%

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {\sin k}^{2}} \]

      9. associate-*r* 72.4%

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{\left(k \cdot \left(k \cdot t\right)\right)} \cdot {\sin k}^{2}} \]

      10. associate-*l* 72.4%

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(\left(k \cdot t\right) \cdot {\sin k}^{2}\right)}} \]

      11. *-commutative 72.4%

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left({\sin k}^{2} \cdot \left(k \cdot t\right)\right)}} \]

   10. Simplified 72.4%

       \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left({\sin k}^{2} \cdot \left(k \cdot t\right)\right)}} \]

   11. Taylor expanded in k around 0 55.9%

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(\color{blue}{{k}^{2}} \cdot \left(k \cdot t\right)\right)} \]
   12. Step-by-step derivation

       1. unpow2 55.9%

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot t\right)\right)} \]

   13. Simplified 55.9%

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot t\right)\right)} \]

   if 8.0000000000000001e85 < k

   1. Initial program 35.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 35.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 35.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 35.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 35.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 35.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 35.3%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 35.3%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 35.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 35.3%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 35.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 58.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 58.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 58.6%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 58.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 58.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 65.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 65.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 65.4%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 55.4%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 55.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]

      2. unpow2 55.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, \frac{{\ell}^{2}}{{k}^{2}}\right)\right) \]

      3. unpow2 55.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right)\right) \]

      4. associate-*r/ 57.8%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \color{blue}{\ell \cdot \frac{\ell}{{k}^{2}}}\right)\right) \]

      5. unpow2 57.8%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right) \]

   9. Simplified 57.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{k \cdot k}\right)}\right) \]

   10. Taylor expanded in k around inf 54.8%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot t}\right)} \]
   11. Step-by-step derivation

       1. times-frac 52.5%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}\right)}\right) \]

       2. unpow2 52.5%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t}\right)\right) \]

       3. unpow2 52.5%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{t}\right)\right) \]

       4. times-frac 68.3%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{t}\right)\right) \]

   12. Simplified 68.3%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-16}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 8 \cdot 10^{+85}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t}\right)\right)\\ \end{array} \]

Alternative 11: 65.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{k \cdot k}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.1e-11)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (* 2.0 (* 0.3333333333333333 (* (* l (/ l t)) (/ (cos k) (* k k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.1e-11) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 * (0.3333333333333333 * ((l * (l / t)) * (cos(k) / (k * k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.1d-11) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 * (0.3333333333333333d0 * ((l * (l / t)) * (cos(k) / (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.1e-11) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * (0.3333333333333333 * ((l * (l / t)) * (Math.cos(k) / (k * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.1e-11:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 * (0.3333333333333333 * ((l * (l / t)) * (math.cos(k) / (k * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.1e-11)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(0.3333333333333333 * Float64(Float64(l * Float64(l / t)) * Float64(cos(k) / Float64(k * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.1e-11)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 * (0.3333333333333333 * ((l * (l / t)) * (cos(k) / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.1e-11], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.3333333333333333 * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1000000000000001e-11

   1. Initial program 61.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 61.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 61.7%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 62.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 62.1%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 61.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 61.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 74.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 74.5%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 74.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 74.6%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 74.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 74.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Taylor expanded in k around 0 53.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   9. Step-by-step derivation

      1. associate-/l/ 53.9%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      2. unpow2 53.9%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}{{k}^{2}} \]

      3. associate-*l/ 58.1%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{{t}^{3}} \cdot \ell}}{{k}^{2}} \]

      4. unpow2 58.1%

         \[\leadsto \frac{\frac{\ell}{{t}^{3}} \cdot \ell}{\color{blue}{k \cdot k}} \]

      5. times-frac 70.1%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

      6. associate-/r* 68.3%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3} \cdot k}} \cdot \frac{\ell}{k} \]

      7. associate-/l/ 71.0%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \cdot \frac{\ell}{k} \]

   10. Simplified 71.0%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{\ell}{k}} \]

   if 1.1000000000000001e-11 < k

   1. Initial program 42.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 42.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 42.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 42.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 42.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 42.6%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 42.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 64.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 64.1%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 64.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 64.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 64.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 68.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 68.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 68.2%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 55.0%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 55.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]

      2. unpow2 55.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, \frac{{\ell}^{2}}{{k}^{2}}\right)\right) \]

      3. unpow2 55.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right)\right) \]

      4. associate-*r/ 56.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \color{blue}{\ell \cdot \frac{\ell}{{k}^{2}}}\right)\right) \]

      5. unpow2 56.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right) \]

   9. Simplified 56.5%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{k \cdot k}\right)}\right) \]

   10. Taylor expanded in l around 0 54.7%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)\right)}{{k}^{2} \cdot t}} \]
   11. Step-by-step derivation

       1. associate-*r* 54.7%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left({\ell}^{2} \cdot \cos k\right) \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}}{{k}^{2} \cdot t} \]

       2. unpow2 54.7%

          \[\leadsto 2 \cdot \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right) \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}{{k}^{2} \cdot t} \]

       3. associate-*l* 54.7%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)} \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}{{k}^{2} \cdot t} \]

       4. unpow2 54.7%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{\color{blue}{k \cdot k}}\right)}{{k}^{2} \cdot t} \]

       5. unpow2 54.7%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

       6. associate-*l* 56.5%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   12. Simplified 56.5%

       \[\leadsto 2 \cdot \color{blue}{\frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{k \cdot \left(k \cdot t\right)}} \]

   13. Taylor expanded in k around inf 53.5%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot t}\right)} \]
   14. Step-by-step derivation

       1. *-commutative 53.5%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{t \cdot {k}^{2}}}\right) \]

       2. times-frac 50.7%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{k}^{2}}\right)}\right) \]

       3. unpow2 50.7%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot \frac{\cos k}{{k}^{2}}\right)\right) \]

       4. associate-*r/ 56.9%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)} \cdot \frac{\cos k}{{k}^{2}}\right)\right) \]

       5. unpow2 56.9%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{\color{blue}{k \cdot k}}\right)\right) \]

   15. Simplified 56.9%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{k \cdot k}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{k \cdot k}\right)\right)\\ \end{array} \]

Alternative 12: 65.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5e-11)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (* 2.0 (* 0.3333333333333333 (* (* (/ l k) (/ l k)) (/ (cos k) t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-11) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 * (0.3333333333333333 * (((l / k) * (l / k)) * (cos(k) / t)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5d-11) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 * (0.3333333333333333d0 * (((l / k) * (l / k)) * (cos(k) / t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-11) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * (0.3333333333333333 * (((l / k) * (l / k)) * (Math.cos(k) / t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 5e-11:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 * (0.3333333333333333 * (((l / k) * (l / k)) * (math.cos(k) / t)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5e-11)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(0.3333333333333333 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5e-11)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 * (0.3333333333333333 * (((l / k) * (l / k)) * (cos(k) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5e-11], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.3333333333333333 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.00000000000000018e-11

   1. Initial program 61.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 61.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 61.7%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 62.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 62.1%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 61.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 61.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 74.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 74.5%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 74.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 74.6%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 74.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 74.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Taylor expanded in k around 0 53.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   9. Step-by-step derivation

      1. associate-/l/ 53.9%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      2. unpow2 53.9%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}{{k}^{2}} \]

      3. associate-*l/ 58.1%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{{t}^{3}} \cdot \ell}}{{k}^{2}} \]

      4. unpow2 58.1%

         \[\leadsto \frac{\frac{\ell}{{t}^{3}} \cdot \ell}{\color{blue}{k \cdot k}} \]

      5. times-frac 70.1%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

      6. associate-/r* 68.3%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3} \cdot k}} \cdot \frac{\ell}{k} \]

      7. associate-/l/ 71.0%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \cdot \frac{\ell}{k} \]

   10. Simplified 71.0%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{\ell}{k}} \]

   if 5.00000000000000018e-11 < k

   1. Initial program 42.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 42.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 42.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 42.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 42.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 42.6%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 42.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 64.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 64.1%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 64.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 64.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 64.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 68.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 68.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 68.2%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 55.0%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 55.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]

      2. unpow2 55.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, \frac{{\ell}^{2}}{{k}^{2}}\right)\right) \]

      3. unpow2 55.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right)\right) \]

      4. associate-*r/ 56.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \color{blue}{\ell \cdot \frac{\ell}{{k}^{2}}}\right)\right) \]

      5. unpow2 56.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right) \]

   9. Simplified 56.5%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{k \cdot k}\right)}\right) \]

   10. Taylor expanded in k around inf 53.5%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot t}\right)} \]
   11. Step-by-step derivation

       1. times-frac 50.7%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}\right)}\right) \]

       2. unpow2 50.7%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t}\right)\right) \]

       3. unpow2 50.7%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{t}\right)\right) \]

       4. times-frac 60.6%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{t}\right)\right) \]

   12. Simplified 60.6%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t}\right)\right)\\ \end{array} \]

Alternative 13: 44.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \left(t \cdot k\right)\\ t_2 := \frac{\ell \cdot \ell}{t_1}\\ \mathbf{if}\;\ell \leq 3.2 \cdot 10^{-203}:\\ \;\;\;\;2 \cdot \left(-0.5 \cdot t_2 + 0.3333333333333333 \cdot t_2\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+213}:\\ \;\;\;\;2 \cdot \frac{\ell}{{k}^{4} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(0.3333333333333333 + \frac{1}{k \cdot k}\right) \cdot \left(\ell \cdot \left(\ell + -0.5 \cdot \left(\ell \cdot \left(k \cdot k\right)\right)\right)\right)}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (* t k))) (t_2 (/ (* l l) t_1)))
   (if (<= l 3.2e-203)
     (* 2.0 (+ (* -0.5 t_2) (* 0.3333333333333333 t_2)))
     (if (<= l 2.1e+213)
       (* 2.0 (/ l (* (pow k 4.0) (/ t l))))
       (*
        2.0
        (/
         (*
          (+ 0.3333333333333333 (/ 1.0 (* k k)))
          (* l (+ l (* -0.5 (* l (* k k))))))
         t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = k * (t * k);
	double t_2 = (l * l) / t_1;
	double tmp;
	if (l <= 3.2e-203) {
		tmp = 2.0 * ((-0.5 * t_2) + (0.3333333333333333 * t_2));
	} else if (l <= 2.1e+213) {
		tmp = 2.0 * (l / (pow(k, 4.0) * (t / l)));
	} else {
		tmp = 2.0 * (((0.3333333333333333 + (1.0 / (k * k))) * (l * (l + (-0.5 * (l * (k * k)))))) / t_1);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = k * (t * k)
    t_2 = (l * l) / t_1
    if (l <= 3.2d-203) then
        tmp = 2.0d0 * (((-0.5d0) * t_2) + (0.3333333333333333d0 * t_2))
    else if (l <= 2.1d+213) then
        tmp = 2.0d0 * (l / ((k ** 4.0d0) * (t / l)))
    else
        tmp = 2.0d0 * (((0.3333333333333333d0 + (1.0d0 / (k * k))) * (l * (l + ((-0.5d0) * (l * (k * k)))))) / t_1)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * (t * k);
	double t_2 = (l * l) / t_1;
	double tmp;
	if (l <= 3.2e-203) {
		tmp = 2.0 * ((-0.5 * t_2) + (0.3333333333333333 * t_2));
	} else if (l <= 2.1e+213) {
		tmp = 2.0 * (l / (Math.pow(k, 4.0) * (t / l)));
	} else {
		tmp = 2.0 * (((0.3333333333333333 + (1.0 / (k * k))) * (l * (l + (-0.5 * (l * (k * k)))))) / t_1);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k * (t * k)
	t_2 = (l * l) / t_1
	tmp = 0
	if l <= 3.2e-203:
		tmp = 2.0 * ((-0.5 * t_2) + (0.3333333333333333 * t_2))
	elif l <= 2.1e+213:
		tmp = 2.0 * (l / (math.pow(k, 4.0) * (t / l)))
	else:
		tmp = 2.0 * (((0.3333333333333333 + (1.0 / (k * k))) * (l * (l + (-0.5 * (l * (k * k)))))) / t_1)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(t * k))
	t_2 = Float64(Float64(l * l) / t_1)
	tmp = 0.0
	if (l <= 3.2e-203)
		tmp = Float64(2.0 * Float64(Float64(-0.5 * t_2) + Float64(0.3333333333333333 * t_2)));
	elseif (l <= 2.1e+213)
		tmp = Float64(2.0 * Float64(l / Float64((k ^ 4.0) * Float64(t / l))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(0.3333333333333333 + Float64(1.0 / Float64(k * k))) * Float64(l * Float64(l + Float64(-0.5 * Float64(l * Float64(k * k)))))) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * (t * k);
	t_2 = (l * l) / t_1;
	tmp = 0.0;
	if (l <= 3.2e-203)
		tmp = 2.0 * ((-0.5 * t_2) + (0.3333333333333333 * t_2));
	elseif (l <= 2.1e+213)
		tmp = 2.0 * (l / ((k ^ 4.0) * (t / l)));
	else
		tmp = 2.0 * (((0.3333333333333333 + (1.0 / (k * k))) * (l * (l + (-0.5 * (l * (k * k)))))) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[l, 3.2e-203], N[(2.0 * N[(N[(-0.5 * t$95$2), $MachinePrecision] + N[(0.3333333333333333 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.1e+213], N[(2.0 * N[(l / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(0.3333333333333333 + N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l + N[(-0.5 * N[(l * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.2e-203

   1. Initial program 59.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 59.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 49.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 49.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 59.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 59.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 59.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 59.3%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 57.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 57.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 61.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 61.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 61.8%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 63.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 63.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 63.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 63.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 63.7%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 34.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
   8. Step-by-step derivation

      1. cancel-sign-sub-inv 34.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]

      2. fma-def 34.7%

         \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      3. unpow2 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      4. unpow2 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      5. associate-*r* 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}}, \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      6. unpow2 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      7. *-commutative 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      8. metadata-eval 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right) + \color{blue}{0.3333333333333333} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      9. unpow2 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right) + 0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}\right) \]

      10. unpow2 34.7%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right) + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

      11. associate-*r* 34.7%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right) + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

   9. Simplified 34.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right) + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right)} \]

   10. Taylor expanded in k around inf 35.9%

       \[\leadsto 2 \cdot \left(\color{blue}{-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right) \]
   11. Step-by-step derivation

       1. unpow2 35.9%

          \[\leadsto 2 \cdot \left(-0.5 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right) \]

       2. unpow2 35.9%

          \[\leadsto 2 \cdot \left(-0.5 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right) \]

       3. associate-*r* 39.8%

          \[\leadsto 2 \cdot \left(-0.5 \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right) \]

   12. Simplified 39.8%

       \[\leadsto 2 \cdot \left(\color{blue}{-0.5 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right) \]

   if 3.2e-203 < l < 2.1000000000000001e213

   1. Initial program 52.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 53.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 51.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 51.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 53.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 53.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 53.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 53.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 54.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 54.1%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 54.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 63.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 63.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 62.9%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 64.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 64.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 68.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 68.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 68.5%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 48.9%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 48.9%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. associate-/l* 52.7%

         \[\leadsto 2 \cdot \color{blue}{\frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]

      3. *-commutative 52.7%

         \[\leadsto 2 \cdot \frac{\ell}{\frac{\color{blue}{t \cdot {k}^{4}}}{\ell}} \]

   9. Simplified 52.7%

      \[\leadsto 2 \cdot \color{blue}{\frac{\ell}{\frac{t \cdot {k}^{4}}{\ell}}} \]

   10. Taylor expanded in t around 0 52.7%

       \[\leadsto 2 \cdot \frac{\ell}{\color{blue}{\frac{{k}^{4} \cdot t}{\ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 52.7%

          \[\leadsto 2 \cdot \frac{\ell}{\frac{\color{blue}{t \cdot {k}^{4}}}{\ell}} \]

       2. associate-*l/ 54.7%

          \[\leadsto 2 \cdot \frac{\ell}{\color{blue}{\frac{t}{\ell} \cdot {k}^{4}}} \]

   12. Simplified 54.7%

       \[\leadsto 2 \cdot \frac{\ell}{\color{blue}{\frac{t}{\ell} \cdot {k}^{4}}} \]

   if 2.1000000000000001e213 < l

   1. Initial program 50.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 50.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 50.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 50.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 50.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 50.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 50.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 50.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 50.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 50.0%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 69.4%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 69.4%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 69.4%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 69.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 69.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 69.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 69.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 69.4%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 69.4%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 69.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]

      2. unpow2 69.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, \frac{{\ell}^{2}}{{k}^{2}}\right)\right) \]

      3. unpow2 69.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right)\right) \]

      4. associate-*r/ 69.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \color{blue}{\ell \cdot \frac{\ell}{{k}^{2}}}\right)\right) \]

      5. unpow2 69.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right) \]

   9. Simplified 69.4%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{k \cdot k}\right)}\right) \]

   10. Taylor expanded in l around 0 69.4%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)\right)}{{k}^{2} \cdot t}} \]
   11. Step-by-step derivation

       1. associate-*r* 69.4%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left({\ell}^{2} \cdot \cos k\right) \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}}{{k}^{2} \cdot t} \]

       2. unpow2 69.4%

          \[\leadsto 2 \cdot \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right) \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}{{k}^{2} \cdot t} \]

       3. associate-*l* 69.4%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)} \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}{{k}^{2} \cdot t} \]

       4. unpow2 69.4%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{\color{blue}{k \cdot k}}\right)}{{k}^{2} \cdot t} \]

       5. unpow2 69.4%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

       6. associate-*l* 69.4%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   12. Simplified 69.4%

       \[\leadsto 2 \cdot \color{blue}{\frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{k \cdot \left(k \cdot t\right)}} \]

   13. Taylor expanded in k around 0 69.3%

       \[\leadsto 2 \cdot \frac{\left(\ell \cdot \color{blue}{\left(\ell + -0.5 \cdot \left({k}^{2} \cdot \ell\right)\right)}\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{k \cdot \left(k \cdot t\right)} \]
   14. Step-by-step derivation

       1. *-commutative 69.3%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell + -0.5 \cdot \color{blue}{\left(\ell \cdot {k}^{2}\right)}\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{k \cdot \left(k \cdot t\right)} \]

       2. unpow2 69.3%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell + -0.5 \cdot \left(\ell \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{k \cdot \left(k \cdot t\right)} \]

   15. Simplified 69.3%

       \[\leadsto 2 \cdot \frac{\left(\ell \cdot \color{blue}{\left(\ell + -0.5 \cdot \left(\ell \cdot \left(k \cdot k\right)\right)\right)}\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{k \cdot \left(k \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.2 \cdot 10^{-203}:\\ \;\;\;\;2 \cdot \left(-0.5 \cdot \frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+213}:\\ \;\;\;\;2 \cdot \frac{\ell}{{k}^{4} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(0.3333333333333333 + \frac{1}{k \cdot k}\right) \cdot \left(\ell \cdot \left(\ell + -0.5 \cdot \left(\ell \cdot \left(k \cdot k\right)\right)\right)\right)}{k \cdot \left(t \cdot k\right)}\\ \end{array} \]

Alternative 14: 58.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell}{\frac{t \cdot {k}^{4}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.6e-37)
   (* (/ l (* k k)) (/ l (pow t 3.0)))
   (* 2.0 (/ l (/ (* t (pow k 4.0)) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.6e-37) {
		tmp = (l / (k * k)) * (l / pow(t, 3.0));
	} else {
		tmp = 2.0 * (l / ((t * pow(k, 4.0)) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.6d-37) then
        tmp = (l / (k * k)) * (l / (t ** 3.0d0))
    else
        tmp = 2.0d0 * (l / ((t * (k ** 4.0d0)) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.6e-37) {
		tmp = (l / (k * k)) * (l / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * (l / ((t * Math.pow(k, 4.0)) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.6e-37:
		tmp = (l / (k * k)) * (l / math.pow(t, 3.0))
	else:
		tmp = 2.0 * (l / ((t * math.pow(k, 4.0)) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.6e-37)
		tmp = Float64(Float64(l / Float64(k * k)) * Float64(l / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(l / Float64(Float64(t * (k ^ 4.0)) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.6e-37)
		tmp = (l / (k * k)) * (l / (t ^ 3.0));
	else
		tmp = 2.0 * (l / ((t * (k ^ 4.0)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.6e-37], N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l / N[(N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.5999999999999998e-37

   1. Initial program 61.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.2%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 61.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 62.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 61.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 61.3%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 61.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 53.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 53.9%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 53.9%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 57.6%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 57.6%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 57.6%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]

   if 2.5999999999999998e-37 < k

   1. Initial program 42.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 42.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 42.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 42.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 42.8%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 42.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 42.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 62.6%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 62.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 62.5%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 62.5%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 62.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 66.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 66.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 66.5%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 48.0%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 48.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. associate-/l* 54.0%

         \[\leadsto 2 \cdot \color{blue}{\frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]

      3. *-commutative 54.0%

         \[\leadsto 2 \cdot \frac{\ell}{\frac{\color{blue}{t \cdot {k}^{4}}}{\ell}} \]

   9. Simplified 54.0%

      \[\leadsto 2 \cdot \color{blue}{\frac{\ell}{\frac{t \cdot {k}^{4}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell}{\frac{t \cdot {k}^{4}}{\ell}}\\ \end{array} \]

Alternative 15: 64.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-37}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell}{\frac{t \cdot {k}^{4}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.1e-37)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (* 2.0 (/ l (/ (* t (pow k 4.0)) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.1e-37) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 * (l / ((t * pow(k, 4.0)) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.1d-37) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 * (l / ((t * (k ** 4.0d0)) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.1e-37) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * (l / ((t * Math.pow(k, 4.0)) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.1e-37:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 * (l / ((t * math.pow(k, 4.0)) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.1e-37)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(l / Float64(Float64(t * (k ^ 4.0)) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.1e-37)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 * (l / ((t * (k ^ 4.0)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.1e-37], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l / N[(N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.1000000000000001e-37

   1. Initial program 61.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.2%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 61.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 61.9%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 62.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 62.2%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 61.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 61.8%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 74.8%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 74.8%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 74.8%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 74.9%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 74.8%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 74.8%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Taylor expanded in k around 0 53.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   9. Step-by-step derivation

      1. associate-/l/ 53.9%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      2. unpow2 53.9%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}{{k}^{2}} \]

      3. associate-*l/ 57.6%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{{t}^{3}} \cdot \ell}}{{k}^{2}} \]

      4. unpow2 57.6%

         \[\leadsto \frac{\frac{\ell}{{t}^{3}} \cdot \ell}{\color{blue}{k \cdot k}} \]

      5. times-frac 69.8%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

      6. associate-/r* 67.9%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3} \cdot k}} \cdot \frac{\ell}{k} \]

      7. associate-/l/ 70.7%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \cdot \frac{\ell}{k} \]

   10. Simplified 70.7%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{\ell}{k}} \]

   if 2.1000000000000001e-37 < k

   1. Initial program 42.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 42.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 42.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 42.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 42.8%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 42.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 42.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 62.6%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 62.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 62.5%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 62.5%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 62.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 66.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 66.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 66.5%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 48.0%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 48.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. associate-/l* 54.0%

         \[\leadsto 2 \cdot \color{blue}{\frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]

      3. *-commutative 54.0%

         \[\leadsto 2 \cdot \frac{\ell}{\frac{\color{blue}{t \cdot {k}^{4}}}{\ell}} \]

   9. Simplified 54.0%

      \[\leadsto 2 \cdot \color{blue}{\frac{\ell}{\frac{t \cdot {k}^{4}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-37}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell}{\frac{t \cdot {k}^{4}}{\ell}}\\ \end{array} \]

Alternative 16: 44.5% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \left(t \cdot k\right)\\ t_2 := \frac{\ell \cdot \ell}{t_1}\\ \mathbf{if}\;\ell \leq 4.9 \cdot 10^{-203}:\\ \;\;\;\;2 \cdot \left(-0.5 \cdot t_2 + 0.3333333333333333 \cdot t_2\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+213}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(0.3333333333333333 + \frac{1}{k \cdot k}\right) \cdot \left(\ell \cdot \left(\ell + -0.5 \cdot \left(\ell \cdot \left(k \cdot k\right)\right)\right)\right)}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (* t k))) (t_2 (/ (* l l) t_1)))
   (if (<= l 4.9e-203)
     (* 2.0 (+ (* -0.5 t_2) (* 0.3333333333333333 t_2)))
     (if (<= l 2.7e+213)
       (* 2.0 (/ (* (/ l k) (/ l k)) t_1))
       (*
        2.0
        (/
         (*
          (+ 0.3333333333333333 (/ 1.0 (* k k)))
          (* l (+ l (* -0.5 (* l (* k k))))))
         t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = k * (t * k);
	double t_2 = (l * l) / t_1;
	double tmp;
	if (l <= 4.9e-203) {
		tmp = 2.0 * ((-0.5 * t_2) + (0.3333333333333333 * t_2));
	} else if (l <= 2.7e+213) {
		tmp = 2.0 * (((l / k) * (l / k)) / t_1);
	} else {
		tmp = 2.0 * (((0.3333333333333333 + (1.0 / (k * k))) * (l * (l + (-0.5 * (l * (k * k)))))) / t_1);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = k * (t * k)
    t_2 = (l * l) / t_1
    if (l <= 4.9d-203) then
        tmp = 2.0d0 * (((-0.5d0) * t_2) + (0.3333333333333333d0 * t_2))
    else if (l <= 2.7d+213) then
        tmp = 2.0d0 * (((l / k) * (l / k)) / t_1)
    else
        tmp = 2.0d0 * (((0.3333333333333333d0 + (1.0d0 / (k * k))) * (l * (l + ((-0.5d0) * (l * (k * k)))))) / t_1)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * (t * k);
	double t_2 = (l * l) / t_1;
	double tmp;
	if (l <= 4.9e-203) {
		tmp = 2.0 * ((-0.5 * t_2) + (0.3333333333333333 * t_2));
	} else if (l <= 2.7e+213) {
		tmp = 2.0 * (((l / k) * (l / k)) / t_1);
	} else {
		tmp = 2.0 * (((0.3333333333333333 + (1.0 / (k * k))) * (l * (l + (-0.5 * (l * (k * k)))))) / t_1);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k * (t * k)
	t_2 = (l * l) / t_1
	tmp = 0
	if l <= 4.9e-203:
		tmp = 2.0 * ((-0.5 * t_2) + (0.3333333333333333 * t_2))
	elif l <= 2.7e+213:
		tmp = 2.0 * (((l / k) * (l / k)) / t_1)
	else:
		tmp = 2.0 * (((0.3333333333333333 + (1.0 / (k * k))) * (l * (l + (-0.5 * (l * (k * k)))))) / t_1)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(t * k))
	t_2 = Float64(Float64(l * l) / t_1)
	tmp = 0.0
	if (l <= 4.9e-203)
		tmp = Float64(2.0 * Float64(Float64(-0.5 * t_2) + Float64(0.3333333333333333 * t_2)));
	elseif (l <= 2.7e+213)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) / t_1));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(0.3333333333333333 + Float64(1.0 / Float64(k * k))) * Float64(l * Float64(l + Float64(-0.5 * Float64(l * Float64(k * k)))))) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * (t * k);
	t_2 = (l * l) / t_1;
	tmp = 0.0;
	if (l <= 4.9e-203)
		tmp = 2.0 * ((-0.5 * t_2) + (0.3333333333333333 * t_2));
	elseif (l <= 2.7e+213)
		tmp = 2.0 * (((l / k) * (l / k)) / t_1);
	else
		tmp = 2.0 * (((0.3333333333333333 + (1.0 / (k * k))) * (l * (l + (-0.5 * (l * (k * k)))))) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[l, 4.9e-203], N[(2.0 * N[(N[(-0.5 * t$95$2), $MachinePrecision] + N[(0.3333333333333333 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.7e+213], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(0.3333333333333333 + N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l + N[(-0.5 * N[(l * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 4.9e-203

   1. Initial program 59.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 59.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 49.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 49.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 59.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 59.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 59.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 59.3%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 57.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 57.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 61.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 61.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 61.8%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 63.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 63.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 63.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 63.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 63.7%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 34.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
   8. Step-by-step derivation

      1. cancel-sign-sub-inv 34.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]

      2. fma-def 34.7%

         \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      3. unpow2 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      4. unpow2 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      5. associate-*r* 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}}, \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      6. unpow2 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      7. *-commutative 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      8. metadata-eval 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right) + \color{blue}{0.3333333333333333} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      9. unpow2 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right) + 0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}\right) \]

      10. unpow2 34.7%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right) + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

      11. associate-*r* 34.7%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right) + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

   9. Simplified 34.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right) + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right)} \]

   10. Taylor expanded in k around inf 35.9%

       \[\leadsto 2 \cdot \left(\color{blue}{-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right) \]
   11. Step-by-step derivation

       1. unpow2 35.9%

          \[\leadsto 2 \cdot \left(-0.5 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right) \]

       2. unpow2 35.9%

          \[\leadsto 2 \cdot \left(-0.5 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right) \]

       3. associate-*r* 39.8%

          \[\leadsto 2 \cdot \left(-0.5 \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right) \]

   12. Simplified 39.8%

       \[\leadsto 2 \cdot \left(\color{blue}{-0.5 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right) \]

   if 4.9e-203 < l < 2.7000000000000001e213

   1. Initial program 52.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 53.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 51.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 51.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 53.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 53.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 53.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 53.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 54.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 54.1%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 54.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 63.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 63.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 62.9%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 64.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 64.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 68.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 68.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 68.5%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 54.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 54.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]

      2. unpow2 54.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, \frac{{\ell}^{2}}{{k}^{2}}\right)\right) \]

      3. unpow2 54.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right)\right) \]

      4. associate-*r/ 54.8%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \color{blue}{\ell \cdot \frac{\ell}{{k}^{2}}}\right)\right) \]

      5. unpow2 54.8%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right) \]

   9. Simplified 54.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{k \cdot k}\right)}\right) \]

   10. Taylor expanded in l around 0 54.2%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)\right)}{{k}^{2} \cdot t}} \]
   11. Step-by-step derivation

       1. associate-*r* 54.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left({\ell}^{2} \cdot \cos k\right) \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}}{{k}^{2} \cdot t} \]

       2. unpow2 54.2%

          \[\leadsto 2 \cdot \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right) \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}{{k}^{2} \cdot t} \]

       3. associate-*l* 54.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)} \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}{{k}^{2} \cdot t} \]

       4. unpow2 54.2%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{\color{blue}{k \cdot k}}\right)}{{k}^{2} \cdot t} \]

       5. unpow2 54.2%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

       6. associate-*l* 54.8%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   12. Simplified 54.8%

       \[\leadsto 2 \cdot \color{blue}{\frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{k \cdot \left(k \cdot t\right)}} \]

   13. Taylor expanded in k around 0 50.1%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}}}}{k \cdot \left(k \cdot t\right)} \]
   14. Step-by-step derivation

       1. unpow2 50.1%

          \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{k \cdot \left(k \cdot t\right)} \]

       2. unpow2 50.1%

          \[\leadsto 2 \cdot \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{k \cdot \left(k \cdot t\right)} \]

       3. times-frac 53.9%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{k \cdot \left(k \cdot t\right)} \]

   15. Simplified 53.9%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{k \cdot \left(k \cdot t\right)} \]

   if 2.7000000000000001e213 < l

   1. Initial program 50.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 50.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 50.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 50.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 50.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 50.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 50.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 50.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 50.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 50.0%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 69.4%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 69.4%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 69.4%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 69.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 69.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 69.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 69.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 69.4%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 69.4%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 69.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]

      2. unpow2 69.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, \frac{{\ell}^{2}}{{k}^{2}}\right)\right) \]

      3. unpow2 69.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right)\right) \]

      4. associate-*r/ 69.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \color{blue}{\ell \cdot \frac{\ell}{{k}^{2}}}\right)\right) \]

      5. unpow2 69.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right) \]

   9. Simplified 69.4%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{k \cdot k}\right)}\right) \]

   10. Taylor expanded in l around 0 69.4%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)\right)}{{k}^{2} \cdot t}} \]
   11. Step-by-step derivation

       1. associate-*r* 69.4%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left({\ell}^{2} \cdot \cos k\right) \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}}{{k}^{2} \cdot t} \]

       2. unpow2 69.4%

          \[\leadsto 2 \cdot \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right) \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}{{k}^{2} \cdot t} \]

       3. associate-*l* 69.4%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)} \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}{{k}^{2} \cdot t} \]

       4. unpow2 69.4%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{\color{blue}{k \cdot k}}\right)}{{k}^{2} \cdot t} \]

       5. unpow2 69.4%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

       6. associate-*l* 69.4%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   12. Simplified 69.4%

       \[\leadsto 2 \cdot \color{blue}{\frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{k \cdot \left(k \cdot t\right)}} \]

   13. Taylor expanded in k around 0 69.3%

       \[\leadsto 2 \cdot \frac{\left(\ell \cdot \color{blue}{\left(\ell + -0.5 \cdot \left({k}^{2} \cdot \ell\right)\right)}\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{k \cdot \left(k \cdot t\right)} \]
   14. Step-by-step derivation

       1. *-commutative 69.3%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell + -0.5 \cdot \color{blue}{\left(\ell \cdot {k}^{2}\right)}\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{k \cdot \left(k \cdot t\right)} \]

       2. unpow2 69.3%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell + -0.5 \cdot \left(\ell \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{k \cdot \left(k \cdot t\right)} \]

   15. Simplified 69.3%

       \[\leadsto 2 \cdot \frac{\left(\ell \cdot \color{blue}{\left(\ell + -0.5 \cdot \left(\ell \cdot \left(k \cdot k\right)\right)\right)}\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{k \cdot \left(k \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.9 \cdot 10^{-203}:\\ \;\;\;\;2 \cdot \left(-0.5 \cdot \frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+213}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(0.3333333333333333 + \frac{1}{k \cdot k}\right) \cdot \left(\ell \cdot \left(\ell + -0.5 \cdot \left(\ell \cdot \left(k \cdot k\right)\right)\right)\right)}{k \cdot \left(t \cdot k\right)}\\ \end{array} \]

Alternative 17: 44.4% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \left(t \cdot k\right)\\ t_2 := \frac{\ell \cdot \ell}{t_1}\\ \mathbf{if}\;\ell \leq 1.7 \cdot 10^{-205}:\\ \;\;\;\;2 \cdot \left(-0.5 \cdot t_2 + 0.3333333333333333 \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (* t k))) (t_2 (/ (* l l) t_1)))
   (if (<= l 1.7e-205)
     (* 2.0 (+ (* -0.5 t_2) (* 0.3333333333333333 t_2)))
     (* 2.0 (/ (* (/ l k) (/ l k)) t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = k * (t * k);
	double t_2 = (l * l) / t_1;
	double tmp;
	if (l <= 1.7e-205) {
		tmp = 2.0 * ((-0.5 * t_2) + (0.3333333333333333 * t_2));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) / t_1);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = k * (t * k)
    t_2 = (l * l) / t_1
    if (l <= 1.7d-205) then
        tmp = 2.0d0 * (((-0.5d0) * t_2) + (0.3333333333333333d0 * t_2))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) / t_1)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * (t * k);
	double t_2 = (l * l) / t_1;
	double tmp;
	if (l <= 1.7e-205) {
		tmp = 2.0 * ((-0.5 * t_2) + (0.3333333333333333 * t_2));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) / t_1);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k * (t * k)
	t_2 = (l * l) / t_1
	tmp = 0
	if l <= 1.7e-205:
		tmp = 2.0 * ((-0.5 * t_2) + (0.3333333333333333 * t_2))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) / t_1)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(t * k))
	t_2 = Float64(Float64(l * l) / t_1)
	tmp = 0.0
	if (l <= 1.7e-205)
		tmp = Float64(2.0 * Float64(Float64(-0.5 * t_2) + Float64(0.3333333333333333 * t_2)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * (t * k);
	t_2 = (l * l) / t_1;
	tmp = 0.0;
	if (l <= 1.7e-205)
		tmp = 2.0 * ((-0.5 * t_2) + (0.3333333333333333 * t_2));
	else
		tmp = 2.0 * (((l / k) * (l / k)) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[l, 1.7e-205], N[(2.0 * N[(N[(-0.5 * t$95$2), $MachinePrecision] + N[(0.3333333333333333 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.7000000000000001e-205

   1. Initial program 59.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 59.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 49.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 49.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 59.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 59.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 59.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 59.3%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 57.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 57.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 61.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 61.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 61.8%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 63.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 63.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 63.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 63.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 63.7%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 34.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
   8. Step-by-step derivation

      1. cancel-sign-sub-inv 34.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]

      2. fma-def 34.7%

         \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      3. unpow2 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      4. unpow2 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      5. associate-*r* 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}}, \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      6. unpow2 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      7. *-commutative 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      8. metadata-eval 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right) + \color{blue}{0.3333333333333333} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      9. unpow2 34.7%

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right) + 0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}\right) \]

      10. unpow2 34.7%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right) + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

      11. associate-*r* 34.7%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right) + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

   9. Simplified 34.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}, \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right) + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right)} \]

   10. Taylor expanded in k around inf 35.9%

       \[\leadsto 2 \cdot \left(\color{blue}{-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right) \]
   11. Step-by-step derivation

       1. unpow2 35.9%

          \[\leadsto 2 \cdot \left(-0.5 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right) \]

       2. unpow2 35.9%

          \[\leadsto 2 \cdot \left(-0.5 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right) \]

       3. associate-*r* 39.8%

          \[\leadsto 2 \cdot \left(-0.5 \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right) \]

   12. Simplified 39.8%

       \[\leadsto 2 \cdot \left(\color{blue}{-0.5 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right) \]

   if 1.7000000000000001e-205 < l

   1. Initial program 52.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 52.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 51.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 51.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 52.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 52.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 52.5%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 52.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 53.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 53.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 53.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 64.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 64.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 63.9%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 64.9%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 64.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 68.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 68.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 68.7%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 57.1%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 57.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]

      2. unpow2 57.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, \frac{{\ell}^{2}}{{k}^{2}}\right)\right) \]

      3. unpow2 57.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right)\right) \]

      4. associate-*r/ 57.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \color{blue}{\ell \cdot \frac{\ell}{{k}^{2}}}\right)\right) \]

      5. unpow2 57.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right) \]

   9. Simplified 57.1%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{k \cdot k}\right)}\right) \]

   10. Taylor expanded in l around 0 56.6%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)\right)}{{k}^{2} \cdot t}} \]
   11. Step-by-step derivation

       1. associate-*r* 56.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left({\ell}^{2} \cdot \cos k\right) \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}}{{k}^{2} \cdot t} \]

       2. unpow2 56.6%

          \[\leadsto 2 \cdot \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right) \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}{{k}^{2} \cdot t} \]

       3. associate-*l* 56.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)} \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}{{k}^{2} \cdot t} \]

       4. unpow2 56.6%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{\color{blue}{k \cdot k}}\right)}{{k}^{2} \cdot t} \]

       5. unpow2 56.6%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

       6. associate-*l* 57.1%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   12. Simplified 57.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{k \cdot \left(k \cdot t\right)}} \]

   13. Taylor expanded in k around 0 51.2%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}}}}{k \cdot \left(k \cdot t\right)} \]
   14. Step-by-step derivation

       1. unpow2 51.2%

          \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{k \cdot \left(k \cdot t\right)} \]

       2. unpow2 51.2%

          \[\leadsto 2 \cdot \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{k \cdot \left(k \cdot t\right)} \]

       3. times-frac 55.4%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{k \cdot \left(k \cdot t\right)} \]

   15. Simplified 55.4%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{k \cdot \left(k \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.7 \cdot 10^{-205}:\\ \;\;\;\;2 \cdot \left(-0.5 \cdot \frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot \left(t \cdot k\right)}\\ \end{array} \]

Alternative 18: 57.2% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot \left(t \cdot k\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (/ (* (/ l k) (/ l k)) (* k (* t k)))))

C

double code(double t, double l, double k) {
	return 2.0 * (((l / k) * (l / k)) / (k * (t * k)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (((l / k) * (l / k)) / (k * (t * k)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (((l / k) * (l / k)) / (k * (t * k)));
}

Python

def code(t, l, k):
	return 2.0 * (((l / k) * (l / k)) / (k * (t * k)))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) / Float64(k * Float64(t * k))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (((l / k) * (l / k)) / (k * (t * k)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 56.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. associate-*l* 50.2%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. sqr-neg 50.2%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. associate-*l* 56.4%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. *-commutative 56.4%

      \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. sqr-neg 56.4%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-*l/ 56.7%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   8. associate-*r/ 56.1%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   9. associate-/r/ 56.1%

      \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

3. Simplified 56.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around inf 62.6%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
5. Step-by-step derivation

   1. *-commutative 62.6%

      \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   2. associate-*r* 62.6%

      \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

   3. times-frac 63.8%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

   4. unpow2 63.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

   5. associate-*l* 65.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

   6. unpow2 65.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

6. Simplified 65.7%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

7. Taylor expanded in k around 0 58.5%

   \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]
8. Step-by-step derivation

   1. fma-def 58.5%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]

   2. unpow2 58.5%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, \frac{{\ell}^{2}}{{k}^{2}}\right)\right) \]

   3. unpow2 58.5%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right)\right) \]

   4. associate-*r/ 58.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \color{blue}{\ell \cdot \frac{\ell}{{k}^{2}}}\right)\right) \]

   5. unpow2 58.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right) \]

9. Simplified 58.8%

   \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{k \cdot k}\right)}\right) \]

10. Taylor expanded in l around 0 57.5%

    \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)\right)}{{k}^{2} \cdot t}} \]
11. Step-by-step derivation

    1. associate-*r* 57.5%

       \[\leadsto 2 \cdot \frac{\color{blue}{\left({\ell}^{2} \cdot \cos k\right) \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}}{{k}^{2} \cdot t} \]

    2. unpow2 57.5%

       \[\leadsto 2 \cdot \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right) \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}{{k}^{2} \cdot t} \]

    3. associate-*l* 57.5%

       \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)} \cdot \left(0.3333333333333333 + \frac{1}{{k}^{2}}\right)}{{k}^{2} \cdot t} \]

    4. unpow2 57.5%

       \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{\color{blue}{k \cdot k}}\right)}{{k}^{2} \cdot t} \]

    5. unpow2 57.5%

       \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

    6. associate-*l* 58.1%

       \[\leadsto 2 \cdot \frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

12. Simplified 58.1%

    \[\leadsto 2 \cdot \color{blue}{\frac{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{k \cdot \left(k \cdot t\right)}} \]

13. Taylor expanded in k around 0 54.3%

    \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}}}}{k \cdot \left(k \cdot t\right)} \]
14. Step-by-step derivation

    1. unpow2 54.3%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{k \cdot \left(k \cdot t\right)} \]

    2. unpow2 54.3%

       \[\leadsto 2 \cdot \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{k \cdot \left(k \cdot t\right)} \]

    3. times-frac 56.3%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{k \cdot \left(k \cdot t\right)} \]

15. Simplified 56.3%

    \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{k \cdot \left(k \cdot t\right)} \]

16. Final simplification 56.3%

    \[\leadsto 2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot \left(t \cdot k\right)} \]

Reproduce

herbie shell --seed 2023275 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))